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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: comp.theory
Subject: =?utf-8?Q?Re:_Tarski_/_G=C3=B6del_and_redefining_the_Foundation_of_Logic?=
Date: Thu, 25 Jul 2024 11:55:43 +0300
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On 2024-07-23 14:53:21 +0000, olcott said:

> On 7/23/2024 3:07 AM, Mikko wrote:
>> On 2024-07-22 14:40:41 +0000, olcott said:
>> 
>>> On 7/22/2024 3:14 AM, Mikko wrote:
>>>> On 2024-07-21 13:20:04 +0000, olcott said:
>>>> 
>>>>> On 7/21/2024 4:27 AM, Mikko wrote:
>>>>>> On 2024-07-20 13:22:31 +0000, olcott said:
>>>>>> 
>>>>>>> On 7/20/2024 3:42 AM, Mikko wrote:
>>>>>>>> On 2024-07-19 13:48:49 +0000, olcott said:
>>>>>>>> 
>>>>>>>>> 
>>>>>>>>> Some undecidable expressions are only undecidable because
>>>>>>>>> they are self contradictory. In other words they are undecidable
>>>>>>>>> because there is something wrong with them.
>>>>>>>> 
>>>>>>>> Being self-contradictory is a semantic property. Being uncdecidable is
>>>>>>>> independent of any semantics.
>>>>>>> 
>>>>>>> Not it is not. When an expression is neither true nor false
>>>>>>> that makes it neither provable nor refutable.
>>>>>> 
>>>>>> There is no aithmetic sentence that is neither true or false. If the sentnece
>>>>>> contains both existentia and universal quantifiers it may be hard to find out
>>>>>> whether it is true or false but there is no sentence that is neither.
>>>>>> 
>>>>>>>  As Richard
>>>>>>> Montague so aptly showed Semantics can be specified syntactically.
>>>>>>> 
>>>>>>>> An arithmetic sentence is always about
>>>>>>>> numbers, not about sentences.
>>>>>>> 
>>>>>>> So when Gödel tried to show it could be about provability
>>>>>>> he was wrong before he even started?
>>>>>> 
>>>>>> Gödel did not try to show that an arithmetic sentence is about provability.
>>>>>> He constructed a sentence about numbers that is either true and provable
>>>>>> or false and unprovable in the theory that is an extension of Peano 
>>>>>> arithmetics.
>>>>>> 
>>>>> 
>>>>> You just directly contradicted yourself.
>>>> 
>>>> I don't, and you cant show any contradiction.
>>>> 
>>> 
>>> Gödel's proof had nothing what-so-ever to do with provability
>>> except that he proved that g is unprovable in PA.
>> 
>> He also proved that its negation is unprovable in PA. He also proved
>> that every consistent extension of PA has a an sentence (different
>> from g) such that both it and its negation are unprovable.
>> 
> 
> L is the language of a formal mathematical system.
> x is an expression of that language.
> 
> When we understand that True(L,x) means that there is a finite
> sequence of truth preserving operations in L from the semantic
> meaning of x to x in L, then mathematical incompleteness is abolished.

No, it is not. From the meaning of "formal mathematical system" follows
that whether x is an expression of language L does not depend on semantics
or L is not a language of a formal mathiematical system. In addition,
the system is incomplete if there is a sentence that can be determined
to be true from the meaning of x but cannot be proven in the system.

-- 
Mikko